There are lots of logic systems (especially modal logic systems)
that are related to each other by the addition of an axiom or two.

I have only been able to find a few summaries of such
interrelationships, and none of those on-line.

What few cross references that I have seen give "facts" but no clue
as to where they came from. If a reference is given, it only points to
a book, and you have no idea where in the book the fact is given. I
suspect that the main contribution that these pages can make is to
provide a cross reference table that one can actually use to go look
up the interesting facts that one may find here.

There are important facts about systems that I have not yet had time to enter,
and there are important facts about systems that I don't happen to
know.

If there is an important fact of a system that you think I should
have listed, and I didn't, please send me that fact and a reference
where I can look up the information.

If you have comments (positive or negative) about these
pages, I would like to hear about them. If you spot errors, I would
like to hear about it. I can be contacted at: John.Halleck@utah.edu If there
are facts you think that need to be mentioned on a system, sending me
mail may get that system moved up the todo list so that it gets done
earlier. This is largely a feedback driven effort.

I have made references as I find them, because of this, I may not
always have a pointer to the "best" references to a system. If you
know better ones than I have here, please contact me.

I think it is time to make the cross references on these pages
automatically generated. It would nice to have machine
readable/processable files for the various axiom sets also.

I can't do these pages in all possible notations, so the
abbreviated notation I try to use here is listed below:

And: &

Or: +

Not: -

Implication: >

Strict implication: =>

Equivalence: ==

Strict equivalence: <=>

Quantification

Universal quantification: forall(x,...)

Existential quantification: exists(x,...)

Modalities

Alethic

Possibility: M

Necessity: L

Deontic

Obligatory: O

Permission: P

Temporal

It will always be: G

It has always been: H

Past: P

Future: F

Doxastic Logic

Bx x believes that ...

Epistemic

Kx it is known that

Provability

Px: x is provable

I will sometimes give other notations also, so you'll sometimes see something
like p>(q>p) [CpCqp]

Different books use different names for axioms and deduction rules.
Therefore there is no one set of names that is going to agree with all
the sources. I have made an effort here to have a consistent naming,
so my naming will not always have the same name as the original
source, although I've made some effort to at least note the original
names.

K. E. Pledger's Enumeration of systems from his 1972 Paper "Modalities of
Systems Containing S3". The article proves
the systems distinct. It also provides a consistent naming, to
avoid confusions (such as Anderson and Åqvist both having a system
called S7.5, and those systems not being equivalent)
R. I. Goldblatt has published a paper [Goldblatt, 1973] "A
Model-Theoretic Study of some Systems Containing S3" that gives
completeness proof's for all the systems in Pledger's enumeration.
The relationships between the main systems are: (This diagram from
[Pledger, 1972, 273],
and is used here by permission of Dr. Pledger.)